Integral Equations and Operator Theory最新文献

In this paper we introduce the concept of matrix-valued q-rational functions. In comparison to the classical case, we give different characterizations with principal emphasis on realizations and discuss algebraic manipulations. We also study the concept of Schur multipliers and complete Nevanlinna–Pick kernels in the context of q-deformed reproducing kernel Hilbert spaces and provide first applications in terms of an interpolation problem using Schur multipliers and complete Nevanlinna–Pick kernels.

This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, ({mathcal {E}}^N) on (L^2(E,mu _N)) for (Nin {mathbb {N}}), in the framework of converging Hilbert spaces by K. Kuwae and T. Shioya. The basic assumption is weak measure convergence of the family ({(mu _N)}_{N}) on the state space E—either a separable Hilbert space or a locally convex topological vector space. Apart from that, the conditions on ({(mu _N)}_{N}) try to impose as little restrictions as possible. The problem has fully been solved if the family ({(mu _N)}_{N}) contain only log-concave measures, due to Ambrosio et al. (Probab Theory Relat. Fields 145:517–564, 2009). However, for a large class of convergence problems the assumption of log-concavity fails. The article suggests a way to overcome this hindrance, as it presents a new approach. Combining the theory of Dirichlet forms with methods from numerical analysis we find abstract criteria for Mosco convergence of standard gradient forms with varying reference measures. These include cases in which the measures are not log-concave. To demonstrate the accessibility of our abstract theory we discuss a first application, generalizing an approximation result by Bounebache and Zambotti (J Theor Probab 27:168–201, 2014).

We consider fermionic ground states of the Landau Hamiltonian, (H_B), in a constant magnetic field of strength (B>0) in ({mathbb {R}}^2) at some fixed Fermi energy (mu >0), described by the Fermi projection (P_B:=1(H_Ble mu )). For some fixed bounded domain (Lambda subset {mathbb {R}}^2) with boundary set (partial Lambda ) and an (L>0) we restrict these ground states spatially to the scaled domain (L Lambda ) and denote the corresponding localised Fermi projection by (P_B(LLambda )). Then we study the scaling of the Hilbert-space trace, (textrm{tr} f(P_B(LLambda ))), for polynomials f with (f(0)=f(1)=0) of these localised ground states in the joint limit (Lrightarrow infty ) and (Brightarrow 0). We obtain to leading order logarithmically enhanced area-laws depending on the size of LB. Roughly speaking, if 1/B tends to infinity faster than L, then we obtain the known enhanced area-law (by the Widom–Sobolev formula) of the form (L ln (L) a(f,mu ) |partial Lambda |) as (Lrightarrow infty ) for the (two-dimensional) Laplacian with Fermi projection (1(H_0le mu )). On the other hand, if L tends to infinity faster than 1/B, then we get an area law with an (L ln (mu /B) a(f,mu ) |partial Lambda |) asymptotic expansion as (Brightarrow 0). The numerical coefficient (a(f,mu )) in both cases is the same and depends solely on the function f and on (mu ). The asymptotic result in the latter case is based upon the recent joint work of Leschke, Sobolev and the second named author [7] for fixed B, a proof of the sine-kernel asymptotics on a global scale, and on the enhanced area-law in dimension one by Landau and Widom. In the special but important case of a quadratic function f we are able to cover the full range of parameters B and L. In general, we have a smaller region of parameters (B, L) where we can prove the two-scale asymptotic expansion (textrm{tr} f(P_B(LLambda ))) as (Lrightarrow infty ) and (Brightarrow 0).

Motivated by current investigations in dilation theory, in both operator theory and operator algebras, and the theory of groupoids, we obtain a generalisation of the Sz-Nagy’s Dilation Theorem for operator valued positive semidefinite maps on (*)-semigroupoids with unit, with varying degrees of aggregation, firstly by (*)-representations with unbounded operators and then we characterise the existence of the corresponding (*)-representations by bounded operators. By linearisation of these constructions, we obtain similar results for operator valued positive semidefinite maps on (*)-algebroids with unit and then, for the special case of (B^*)-algebroids with unit, we obtain a generalisation of the Stinespring’s Dilation Theorem. As an application of the generalisation of the Stinespring’s Dilation Theorem, we show that some natural questions on (C^*)-algebroids are equivalent.

Our initial data is a transfer operator L for a continuous, countable-to-one map (varphi :Delta rightarrow X) defined on an open subset of a locally compact Hausdorff space X. Then L may be identified with a ‘potential’, i.e. a map (varrho :Delta rightarrow X) that need not be continuous unless (varphi ) is a local homeomorphism. We define the crossed product (C_0(X)rtimes L) as a universal (C^*)-algebra with explicit generators and relations, and give an explicit faithful representation of (C_0(X)rtimes L) under which it is generated by weighted composition operators. We explain its relationship with Exel–Royer’s crossed products, quiver (C^*)-algebras of Muhly and Tomforde, (C^*)-algebras associated to complex or self-similar dynamics by Kajiwara and Watatani, and groupoid (C^*)-algebras associated to Deaconu–Renault groupoids. We describe spectra of core subalgebras of (C_0(X)rtimes L), prove uniqueness theorems for (C_0(X)rtimes L) and characterize simplicity of (C_0(X)rtimes L). We give efficient criteria for (C_0(X)rtimes L) to be purely infinite simple and in particular a Kirchberg algebra.

We study the biholomorphic action of the Heisenberg group (mathbb {H}_n) on the Siegel domain (D_{n+1}) ((n ge 1)). Such (mathbb {H}_n)-action allows us to obtain decompositions of both (D_{n+1}) and the weighted Bergman spaces (mathcal {A}^2_lambda (D_{n+1})) ((lambda > -1)). Through the use of symplectic geometry we construct a natural set of coordinates for (D_{n+1}) adapted to (mathbb {H}_n). This yields a useful decomposition of the domain (D_{n+1}). The latter is then used to compute a decomposition of the Bergman spaces (mathcal {A}^2_lambda (D_{n+1})) ((lambda > -1)) as direct integrals of Fock spaces. This effectively shows the existence of an interplay between Bergman spaces and Fock spaces through the Heisenberg group (mathbb {H}_n). As an application, we consider (mathcal {T}^{(lambda )}(L^infty (D_{n+1})^{mathbb {H}_n})) the (C^*)-algebra acting on the weighted Bergman space (mathcal {A}^2_lambda (D_{n+1})) ((lambda > -1)) generated by Toeplitz operators whose symbols belong to (L^infty (D_{n+1})^{mathbb {H}_n}) (essentially bounded and (mathbb {H}_n)-invariant). We prove that (mathcal {T}^{(lambda )}(L^infty (D_{n+1})^{mathbb {H}_n})) is commutative and isomorphic to (textrm{VSO}(mathbb {R}_+)) (very slowly oscillating functions on (mathbb {R}_+)), for every (lambda > -1) and (n ge 1).

For the Schrödinger operator (H=-Delta + V({{textbf{x}}})cdot ), acting in the space (L_2({{textbf{R}}}^d),(dge 3)), necessary and sufficient conditions for semi-boundedness and discreteness of its spectrum are obtained without the assumption that the potential (V({{textbf{x}}})) is bounded below. By reducing the problem to study the existence of regular solutions of the Riccati PDE, the necessary conditions for the discreteness of the spectrum of operator H are obtained under the assumption that it is bounded below. These results are similar to the ones obtained by the author in [26] for the one-dimensional case. Furthermore, sufficient conditions for the semi-boundedness and discreteness of the spectrum of H are obtained in terms of a non-increasing rearrangement, mathematical expectation, and standard deviation from the latter for the positive part (V_+({{textbf{x}}})) of the potential (V({{textbf{x}}})) on compact domains that go to infinity, under certain restrictions for its negative part (V_-({{textbf{x}}})). Choosing optimally the vector field associated with the difference between the potential (V({{textbf{x}}})) and its mathematical expectation on the balls that go to infinity, we obtain a condition for semi-boundedness and discreteness of the spectrum for H in terms of solutions of the Neumann problem for the nonhomogeneous (d/(d-1))-Laplace equation. This type of optimization refers to a divergence constrained transportation problem.

Let (n,mge 1) and (alpha >0). We denote by (mathcal {F}_{alpha ,m}) the m-analytic Bargmann–Segal–Fock space, i.e., the Hilbert space of all m-analytic functions defined on (mathbb {C}^n) and square integrables with respect to the Gaussian weight (exp (-alpha |z|^2)). We study the von Neumann algebra (mathcal {A}) of bounded linear operators acting in (mathcal {F}_{alpha ,m}) and commuting with all “horizontal” Weyl translations, i.e., Weyl unitary operators associated to the elements of (mathbb {R}^n). The reproducing kernel of (mathcal {F}_{1,m}) was computed by Youssfi [Polyanalytic reproducing kernels in (mathbb {C}^n), Complex Anal. Synerg., 2021, 7, 28]. Multiplying the elements of (mathcal {F}_{alpha ,m}) by an appropriate weight, we transform this space into another reproducing kernel Hilbert space whose kernel K is invariant under horizontal translations. Using the well-known Fourier connection between Laguerre and Hermite functions, we compute the Fourier transform of K in the “horizontal direction” and decompose it into the sum of d products of Hermite functions, with (d=left( {begin{array}{c}n+m-1 nend{array}}right) ). Finally, applying the scheme proposed by Herrera-Yañez, Maximenko, Ramos-Vazquez [Translation-invariant operators in reproducing kernel Hilbert spaces, Integr. Equ. Oper. Theory, 2022, 94, 31], we show that (mathcal {F}_{alpha ,m}) is isometrically isomorphic to the space of vector-functions (L^2(mathbb {R}^n)^d), and (mathcal {A}) is isometrically isomorphic to the algebra of matrix-functions (L^infty (mathbb {R}^n)^{dtimes d}).

In this short note, we discuss essential positivity of Toeplitz operators on the Fock space, as motivated by a recent question of Perälä and Virtanen (Proc. Amer. Math. Soc. 151:4807–4815, 2023). We give a proper characterization of essential positivity in terms of limit operators. A conjectured characterization of essential positivity of Perälä and Virtanen is disproven when the assumption of radiality is dropped. Nevertheless, when the symbol of the Toeplitz operator is of vanishing mean oscillation, we show that the conjecture of Perälä and Virtanen holds true, even without radiality.

In this article, we study a commutative Banach algebra structure on the space (L^1(mathbb {R}^{2n})oplus {mathcal {T}}^1), where the ({mathcal {T}}^1) denotes the trace class operators on (L^2(mathbb {R}^{n})). The product of this space is given by the convolutions in quantum harmonic analysis. Towards this goal, we study the closed ideals of this space, and in particular its Gelfand theory. We additionally develop the concept of quantum Segal algebras as an analogue of Segal algebras. We prove that many of the properties of Segal algebras have transfers to quantum Segal algebras. However, it should be noted that in contrast to Segal algebras, quantum Segal algebras are not ideals of the ambient space. We also give examples of different constructions that yield quantum Segal algebras.